Problem 1.1 in Claes Johnson's book "Numerical solution of partial diff. eqs..."

Hallo all,

I'm new to the forum, so I apologize if I am in the wrong section!

I'm studying Claes Johnson's book "Numerical solution of partial differential equations by the Finite Element Method", and I'm trying to solve the proposed problems.

What is not very encouraging is that I'm already stuck at problem 1.1. :-) It seems easy and I understand it intuitively, but I cannot prove it formally. Anyone can help? This would also give me a hint on a methodology and "way of thinking" to solve the next ones.

So, the problem is the following. Let V be the space of functions v continuous on [0,1], with v' piecewise continuous and bounded on [0,1], and v(0) = v(1) = 0.

Demonstrate that, if a function w is continuous on [0,1], and the definite integral between 0 and 1 of w*v is zero for each v belonging to V, then w = 0 on all [0,1].

Can anyone give me a hint?

Thanks a lot!

Ingwer

Re: Problem 1.1 in Claes Johnson's book "Numerical solution of partial diff. eqs..."

It is not hard to solve, only bit tricky.

A proof by contradiction. Suppose that $\displaystyle w\neq0$ over an interval $\displaystyle I$, and without loss of generality let $\displaystyle w>0$.

Construct (as an exercise) a smooth function $\displaystyle v\in V$ such that $\displaystyle v\equiv -1$ at the middle third $\displaystyle J$ of the interval $\displaystyle I$, $\displaystyle v$ ascends to zero on the other two intervals, and $\displaystyle v\equiv0$ outside of $\displaystyle I$. Then we have

$\displaystyle 0=\bigg|\int_I wv\bigg|=\int_I w(-v)\geq \int_J w$

a contradiction since $\displaystyle w>0$ throughout $\displaystyle J$.

Re: Problem 1.1 in Claes Johnson's book "Numerical solution of partial diff. eqs..."

Re: Problem 1.1 in Claes Johnson's book "Numerical solution of partial diff. eqs..."

Can someone offer an alternative proof? It seems there has to be a more simple version of this?

Re: Problem 1.1 in Claes Johnson's book "Numerical solution of partial diff. eqs..."

Hello,

Did anyone ever answer your question about problem 1.1 Do you have a solution to 1.2. I started looking at the same book.